Question: Simplify and expand the following expression: $ \dfrac{a}{2a - 5}-\dfrac{5a + 5}{5a + 5} $
Solution: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(2a - 5)(5a + 5)$ Multiply the first term by $\dfrac{5a + 5}{5a + 5}$ $ \begin{align*} \dfrac{a}{2a - 5} \times \dfrac{5a + 5}{5a + 5} & = \dfrac{(a)(5a + 5)}{(2a - 5)(5a + 5)} \\ & = \dfrac{5a^2 + 5a}{(2a - 5)(5a + 5)}\end{align*} $ Multiply the second term by $\dfrac{2a - 5}{2a - 5}$ $ \begin{align*} \dfrac{5a + 5}{5a + 5} \times \dfrac{2a - 5}{2a - 5} & = \dfrac{(5a + 5)(2a - 5)}{(5a + 5)(2a - 5)} \\ & = \dfrac{10a^2 - 15a - 25}{(5a + 5)(2a - 5)}\end{align*} $ Now we have: $ = \dfrac{5a^2 + 5a}{(2a - 5)(5a + 5)} - \dfrac{10a^2 - 15a - 25}{(5a + 5)(2a - 5)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{5a^2 + 5a - (10a^2 - 15a - 25)}{(2a - 5)(5a + 5)} $ $ = \dfrac{5a^2 + 5a - 10a^2 + 15a + 25}{(2a - 5)(5a + 5)} $ $ = \dfrac{-5a^2 + 20a + 25}{(2a - 5)(5a + 5)}$ Expand the denominator: $ = \dfrac{-5a^2 + 20a + 25}{10a^2 - 15a - 25}$ Simplify: $ = \dfrac{-a^2 + 4a + 5}{2a^2 - 3a - 5}$